Theorem bnj132 35223

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj132.1	⊢ (𝜑 ↔ ∃𝑥(𝜓 → 𝜒))

Assertion

Ref	Expression
bnj132	⊢ (𝜑 ↔ (𝜓 → ∃𝑥𝜒))

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)

Proof of Theorem bnj132

Step	Hyp	Ref	Expression
1		bnj132.1	. 2 ⊢ (𝜑 ↔ ∃𝑥(𝜓 → 𝜒))
2		19.37v 2030	. 2 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (𝜓 → ∃𝑥𝜒))
3	1, 2	bitri 278	1 ⊢ (𝜑 ↔ (𝜓 → ∃𝑥𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bnj996  35452